Homoclinic orbit expansion of arbitrary trajectories in chaotic systems: classical action function and its memory

Abstract

Special subsets of orbits in chaotic systems, e.g. periodic orbits, heteroclinic orbits, closed orbits, can be considered as skeletons or scaffolds upon which the full dynamics of the system is built. In particular, as demonstrated in previous publications [Phys. Rev. E 95, 062224 (2017), Phys. Rev. E 97, 022216 (2018)], the determination of homoclinic orbits is sufficient for the exact calculation of classical action functions of unstable periodic orbits, which have potential applications in semiclassical trace formulas. Here this previous work is generalized to the calculation of classical action functions of arbitrary trajectory segments in multidimensional chaotic Hamiltonian systems. The unstable trajectory segments' actions are expanded into linear combinations of homoclinic orbit actions that shadow them in a piece-wise fashion. The results lend themselves to an approximation with controllable exponentially small errors, and which demonstrates an exponentially rapid loss of memory of a segment's classical action to its past and future. Furthermore, it does not require an actual construction of the trajectory segment, only its Markov partition sequence. An alternative point of view is also proposed which partitions the trajectories into short segments of transient visits to the neighborhoods of successive periodic orbits, giving rise to a periodic orbit expansion scheme which is equivalent to the homoclinic orbit expansion. This clearly demonstrates that homoclinic and periodic orbits are equally valid skeletal structures for the tessellation of phase-space dynamics.